. Author manuscript; available in PMC: 2020 Jun 14.

Published in final edited form as: Biometrics. 2019 Nov 11;76(2):484–495. doi: 10.1111/biom.13162

Table 2.

An example showing that IPW-S approach, in general, cannot be used for partial testing when covariates are correlated and under violation of positivity. UNADJ - the naive use of the Cox regression that ignores truncation. The correlated covariates are $Z_{1} ~ N (0, 1), P (Z_{2} = 0 | Z_{1} < 0) = 2 / 3 = P (Z_{2} = 1 | Z_{1} > 0) . b i a s (\hat{β})$ is the bias of estimates $\hat{β} .$ SD is defined as in Table 1. SE_a is the analytical estimate of SE. size is the size of 0.05% test for H₀ : β₂ = 0 based on normal approximation and SE_a. The results are based on 1000 replications. *₋ results for IPW-SA are based on 200 replications. r₍₁₀₀₀₎ is an n^th-order statistic of $R_{1}^{*}, \dots, R_{n}^{*}$ in the samples with n = 1000.

	β₁ = 2				β₂ = 0
No violation of positivity assumption: R ~ Gamma(1.4, 0.27), P(T > R) = 0.15, average P(T > r₍₁₀₀₀₎) = 0.0005
	n	$bias ({\hat{β}}_{1})$	SD	SE_a	$bias ({\hat{β}}_{2})$	SD	SE_a	size
IPW-S	100	0.029	0.218	0.209	0.008	0.244	0.227	0.072
	300	0.013	0.115	0.116	−0.001	0.134	0.130	0.048
	500	0.010	0.092	0.090	0.004	0.107	0.101	0.062
	1000	0.001	0.065	0.064	−0.001	0.070	0.072	0.048
UNADJ	1000	−0.130	0.061	0.059	0.000	0.065	0.066	0.042
Light violation of positivity assumption: R ~ Unif[0, 1.5], P(T > R) = 0.562, average P(T > r₍₁₀₀₀₎) = 0.28, a₀ = 0.11
	n	$bias ({\hat{β}}_{1})$	SD	SE_a	$bias ({\hat{β}}_{2})$	SD	SE_a	size
IPW-S	100	−0.263	0.246	0.225	−0.054	0.323	0.300	0.063
	300	−0.277	0.137	0.133	−0.046	0.201	0.189	0.069
	500	−0.284	0.108	0.104	−0.052	0.151	0.149	0.060
	1000	−0.286	0.080	0.076	−0.050	0.123	0.122	0.101
UNADJ	1000	−0.609	0.057	0.058	−0.063	0.074	0.068	0.173
Heavy violation of positivity assumption: R ~ Unif [0, 0.8], P(T > R) = 0.73, P(T > r₍₁₀₀₀₎) = 0.49, a₀ = 0.527
	n	$bias ({\hat{β}}_{1})$	SD	SE_a	$bias ({\hat{β}}_{2})$	SD	SE_a	size
IPW-S	100	−0.539	0.274	0.287	−0.083	0.381	0.400	0.063
	300	−0.545	0.158	0.160	−0.086	0.233	0.223	0.065
	500	−0.554	0.124	0.133	−0.077	0.195	0.180	0.084
	1000	−0.551	0.091	0.092	−0.078	0.143	0.135	0.100
UNADJ	1000	−0.911	0.056	0.057	−0.070	0.071	0.069	0.176
*Under heavy violation of positivity using true a₀ = 0.527
	n	$bias ({\hat{β}}_{1})$	SD		$bias ({\hat{β}}_{2})$	SD
IPW-SA	100	0.023	0.297		−0.055	0.430
	300	0.003	0.148		−0.040	0.272
	500	0.009	0.124		0.001	0.216
	1000	0.003	0.087		0.001	0.164